Electron. J. Differential Equations, Vol. 2017 (2017), No. 132, pp. 1-10.

Existence and nonexistence of solutions for sublinear problems with prescribed number of zeros on exterior domains

Janak Joshi

We prove existence of radial solutions of $\Delta u+K(r)f(u)=0$ on the exterior of the ball, of radius R, centered at the origin in $\mathbb{R}^{N}$ such that $\lim_{r \to \infty} u(r)=0$ if R>0 is sufficiently small. We assume $f:\mathbb{R} \to \mathbb{R}$ is odd and there exists a $\beta>0$ with f<0 on $(0,\beta)$, f>0 on $(\beta,\infty)$ with f sublinear for large u, and $K(r) \sim r^{-\alpha}$ for large r with $ \alpha >2(N-1)$. We also prove nonexistence if R>0 is sufficiently large.

Submitted March 6, 2017. Published May 16, 2017.
Math Subject Classifications: 34B40, 35B05.
Key Words: Exterior domain; sublinear; radial solution.

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Janak Joshi
Department of Mathematics
University of North Texas
Denton, TX 76203, USA
email: janakrajjoshi@my.unt.edu

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